I bought this "phase tester" tool just because it was ridiculously cheap (like $9). You play a burst tone that was supplied on a CD and then hold the device up to the driver under test and it will light a green or red light to tell you the "phase". It seems to work in that regard.

Anyway, the first thing I tested it on is my Phase Technology PC80 2-ways. It showed that the tweeter and woofer are out of phase. I checked the internal wiring and the colors match at least.

This probably shows my complete ignorance of speaker design, but is this phase mismatch intentional by the designer? Is this tool even a useful device or just a gimmick?

Don't confuse phase with polarity.
Polarity is absolute. Polarity can be visually inspected and physically changed by reversing wires.
Phase is relative. Phase is affected by crossovers and distance (among other things). Two drivers are in phase at some frequencies and out of phase at others and a blend of both depending on where you measure. Room reflections and baffle diffraction will also play a role.

When you place the device in front of the tweeter you still measure the woofer but with different path lengths. You also measure diffraction effects. Also, distance from the driver will matter.

If the device is equidistant from both drivers and the noise burst is around the crossover point then it may be useful. Hard to say without seeing it and the test.

The tester needs to be really close to the driver to work (depends on sound level), so interference from the other driver isn't likely. I rechecked by disconnecting one driver at a time and the results are the same.

I really don't see a purpose for this device now that I think about it. Telling phase between say right/left channels is pretty easy to do with just your ears.

It depends on the crossover. It's pretty common for the tweeter to be connected in reverse polarity. If there is a 90 deg phase shift in each driver at the crossover point, what's the best way to get them back in phase again?

I agree that I wouldn't use a 9V on anything except large subwoofers. An AA battery is sufficient for cone woofers and midrange drivers. Trying to see a tweeter dome move with a battery would probably be the last thing you do with the tweeter!

All that said, we mostly care about the interaction of the tweeter and midrange at the crossover frequency. studying the crossover's effect on phase and predicting phase shift is certainly a viable option.